Jimmy Carter (term: 1977-1981) oversaw the Iran Hostage Crisis, which occurred in 1979 and lasted over 400 days. It was a massively public situation that had a ton of angles to it, and many argue that this cost Jimmy Carter a second term in office. 

Iran was facing a revolution, and the Iranian students that held the Americans hostage in the American embassy were supported by the powers that were not backed by the American government.

President Ronald Reagan (term: 1981-1989) sealed an executive agreement with USSR general secretary Mikhail Gorbachev that removed certain close-range nuclear weaponry from government's weapons caches.  

Executive agreements do not need Senate approval, but can lay the groundwork for official treaties and other diplomatic progress.  

There are 538 electors, which correspond to the 100 members of the Senate, 435 members of the House of Representatives, and 3 additional members for the District of Columbia. In order to carry the majority of the electoral votes, a President must win at least 270 electoral votes.

A line-item veto allows the President to block parts of a provisional bill in the Legislature, without having to block the whole of the bill. Although the line-item veto has been proposed numerous times in Congress, it has never been approved by both Congress and the Supreme Court; (the Court ruled it unconstitutional in 1998). Presumably, the Legislative and Judicial Branches fear extending too much power to the Executive to control the exact direction of legislative action.

In the event that the electoral college fails to elect a President then the responsibility for choosing a President falls to the House of Representatives. This happened in the election of 1800, when flaws in the original Constitution saw all members of the Democratic-Republican Party casting their votes for Thomas Jefferson and Aaron Burr, thus creating an equal vote for President and Vice-President for both men. This was rectified in 1804 with the passage of the Twelfth Amendment, which stipulated that the electoral college must clearly cast their votes individually for President and Vice-President. It also happened in the election of 1824, when John Quincy Adams was chosen by the House (and Henry Clay) after receiving less of the popular vote and the electoral vote than Andrew Jackson—the so-called "Corrupt Bargain."

An Executive Order allows the President to pass a law outside of Congress. It allows the President to bypass Congress in making laws. It was originally uncommon in American political behavior, but it has gained prominence since the Presidency of Ulysses S. Grant. The usage of Executive Order peaked with Franklin D. Roosevelt and served as a function of his massive overhaul of social security and welfare in America during the New Deal program.

A line-item veto exists in many countries as means for the Executive to veto a small portion of bill without vetoing the entire package; however, in the United States it has consistently either not been supported by Congress or has been ruled unconstitutional by the courts. The President in the United States cannot use a line-item veto.

William Henry Harrison became the first President to die in office, in 1841, and his death left a great deal of confusion about who should replace him as President. The vague language of the Constitution made it confusing as to whether the office of President should immediately devolve to the Vice-President, or whether the Vice-President should simply fulfill the responsibilities of President without the office or title. John Tyler, Harrison’s Vice-President, had himself inaugurated as President and acted quickly to ensure that his position as President could not be questioned, establishing the Tyler Precedent.

A candidate has to be a natural born citizen or a citizen of the United States at the time of the adoption of the Constitution, must be at least 35 years of age, and must have been a resident of the United States for at least 14 years. There is no requirement that a candidate be born in one of the 50 States. The requirement of a natural born citizen is widely interpreted to mean someone who is a citizen by virtue of their birth (e.g. to an American citizen parent), which does not necessarily have to occur in one of the 50 states. For example, John McCain could have been elected President in 2008 despite being born in the Panama Canal Zone.

The Constitution explicitly lays out the procedure for treaty making. After a treaty is negotiated by the President, it is sent to the Senate for approval.  of the Senate must vote yea for the Treaty to be ratified by the United States. A simple majority of the senate is not sufficient to approve a treaty.